356 



THE BELL SYSTEM TECHNICAL JOURNAL, APRIL 1951 



For these values we find unit increments in q at the zeros (negative steps) 

 0,rb25°.47,±55°.76; and at the poles (positive steps) =tl20°.65, ±142°.89, 

 ±158°.99 and =tl73°.17. These five zeros and eight poles on the unit 

 circle in the w-plane are now mapped back to the corresponding points on 

 the ellipse in the ^-plane, where they give the location of the zeros and 

 poles of the approximate transmission function. Figure 21 illustrates the 

 accuracy of the resulting approximation to the prescribed gain and phase. 



1.0 

 0.9 

 0.8 

 0.7 



0.6 



z 

 > 

 >0.5 



D 

 >° 



— 0.^ 

 0-3 

 0.2 

 0.1 



20 



14 



to 

 < 



< 



10 z 



0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 

 FREQUENCY, a; 



Fig. 21 — Gain and phase curves for the Gaussian filter. 

 Example 2. The Coaxial Cable Equalizer 



A section of coaxial line of finite conductivity has an insertion loss pro- 

 portional to \/co- The problem is to design a network which will equalize 

 this distortion, that is, a network which has a transmission function 



F,{p) = kVp 



in the frequency range \o)\ < 1. 



This example is included partly because of its engineering importance, 

 but also because it gives us the opportunity to introduce a particular type 

 of contour, the equipotenlial contour. This consists of fitting the contour C 

 to an equipotenlial of the transmission function, except for an arc at infinity 

 (if C were everywhere equipotential Fi{p) could only be constant). Thus 

 the contour is not closed in the finite part of the plane, but is supposed to 

 be closed through an arc at infinity so chosen that the charges on this arc 

 will not produce any appreciable eflfect in the finite part of the plane. 



